Predictive Inference Based on Markov Chain Monte Carlo Output

TL;DR

This paper analyzes how to evaluate and improve probabilistic forecasts derived from MCMC samples in Bayesian inference, emphasizing the importance of scoring rules and providing asymptotic guarantees for different approximation methods.

Contribution

It introduces a systematic framework for assessing the consistency of predictive distributions from MCMC output using proper scoring rules, with new theoretical results and practical guidelines.

Findings

CRPS provides consistent approximations under minimal assumptions.

Mixture-of-parameters methods perform well in practice.

Logarithmic scoring rule requires stricter conditions.

Abstract

In Bayesian inference, predictive distributions are typically in the form of samples generated via Markov chain Monte Carlo (MCMC) or related algorithms. In this paper, we conduct a systematic analysis of how to make and evaluate probabilistic forecasts from such simulation output. Based on proper scoring rules, we develop a notion of consistency that allows to assess the adequacy of methods for estimating the stationary distribution underlying the simulation output. We then provide asymptotic results that account for the salient features of Bayesian posterior simulators, and derive conditions under which choices from the literature satisfy our notion of consistency. Importantly, these conditions depend on the scoring rule being used, such that the choices of approximation method and scoring rule are intertwined. While the logarithmic rule requires fairly stringent conditions, the continuous ranked probability score (CRPS) yields consistent approximations under minimal assumptions. These results are illustrated in a simulation study and an economic data example. Overall, mixture-of-parameters approximations which exploit the parametric structure of Bayesian models perform particularly well. Under the CRPS, the empirical distribution function is a simple and appealing alternative option.